An Explicit Formula and a Recurrence for OEIS A265233
A counting problem, a Franel-number formula, and a proof of a conjectured recurrence.
Central South University
Juntao Jin.
I work on counting problems and study algebra through its connections with geometry, topology, and arithmetic.
How I work through a question
Recorded questions, exact examples, and the proofs behind them.
Study note · exact algebra calculation
Read the recordFive points, two tangent spaces
An exact comparison of two Artinian quotients with Hilbert function (1, 3, 1).
Study note · chain calculation
Read the recordWhere characteristic enters a resolution
From a six-vertex projective plane to two different Betti numbers.
Study note · classical criterion
Read the recordTwo negative Pell equations that coarse data cannot separate
145 and 305: matching congruence signatures, opposite integral answers.
Study note · worked proof
Read the recordThe relations of (ab, bc, ca)
My question about “rank-two freeness” leads to a relation module, an exact resolution, and three ways to check it.
Research proposals
Three directions, each with a defined first task and explicit known boundaries.
Research proposal
When finite schemes deform
Compare full embedded tangent spaces along an explicit flat family whose starting quotients share Hilbert function (1,3,1), then compute second-order compatibility equations.
Read the project briefResearch proposal
Torsion in the resolution
Fix iterated barycentric subdivision, without adding a disjoint monomial, and compare exact Betti data over Q and F₂. Begin with r=0,1 and h=2,3.
Read the project briefResearch proposal
Beyond quadratic residues
Fix prime q≡5 mod 8 with (5/q)=1. Compare (5/q)₄ and (q/5)₄ with exact Pell periods, then use the norm-sign question as the entry into narrow class groups.
Read the project briefReading in context
From Artinian structure to deformation, from relations to topology. Each book links to the records where its tools appear.
Browse the reading shelfBeyond mathematics
A place for photographs and everyday stories as they become available.
Personal lifeConversations are welcome.
A proof, a reading suggestion, or a mathematical question.